Spherical Parametrization and Remeshingemilp/papers/SphereParam.pdf · regular 2D grid. The process involves ... their composition provides a map I ... to a great circle arc, and

Spherical Parametrization and Remeshing Emil Praun Hugues Hoppe

University of Utah Microsoft Research

original spherical parametrization octahedral parametrization geometry image (lit) remeshed geometry

Figure 1 Demonstration of spherical parametrization and subsequent resampling into a geometry image

Abstract Recently Gu et al [2002] introduced geometry images in which geometry is resampled into a completely regular 2D grid The process involves cutting the surface into a disk using a network of cut paths and then mapping the boundary of this disk to a square Both geometry and other signals are stored as 2D grids with grid samples in implicit correspondence obviating the need to store a parametrization Also the boundary pa-rametrization makes both geometry and textures seamless

The traditional approach for parametrizing a surface involves cutting it into charts and mapping these piecewise onto a planar domain We introduce a robust technique for directly parametriz-ing a genus-zero surface onto a spherical domain A key ingredient for making such a parametrization practical is the minimization of a stretch-based measure to reduce scale-distortion and thereby prevent undersampling Our second contri-bution is a scheme for sampling the spherical domain using uniformly subdivided polyhedral domains namely the tetrahe-dron octahedron and cube We show that these particular semi-regular samplings can be conveniently represented as completely regular 2D grids ie geometry images Moreover these images have simple boundary extension rules that aid many processing operations Applications include geometry remeshing level-of-detail morphing compression and smooth surface subdivision

In all three approaches the surface is first cut into one or more disk-like charts using a network of cut paths and a parametriza-tion is formed piecewise on each chart The a priori construction of the chart boundaries or cut paths is heuristic and constrains the quality of the attainable parametrization In texture atlases both the number of charts and their surface extents are selected heuris-tically to minimize parametric distortion onto planar polygons while also maintaining good packing efficiency In semi-regular remeshing surface charts are selected to have low-distortion maps onto regular domain faces and to have approximately the same size Finally in geometry images the surface is heuristically cut into a disk that hopefully maps well onto a square

Keywords texture mapping remeshing geometry images meshes

1 Introduction Surface parametrization To associate a given surface with a planar domain the traditional approach is to partition the surface into charts parametrize these in the plane and pack them into a texture atlas as shown in the example on the right [eg

] One main drawback of an atlas is the

presence of visible seams on the surface Shefftry to hide these se in high-curvature regions An altern

Sorkine et al [ ] take the interesting approach of parametriz-ing a chart during its incremental growth to bound distortion Thus the creation of cut paths is guided by the parametrization

2002 Maillot et al 1993 Cignoni et

al 1998 Sander et al 2001 Leacutevy et al 2002 In this paper we construct for a common class of models a con-

tinuous unconstrained parametrization without any cutting er and Hart [2002]

ative is semi-regular remeshing whereby ams Spherical parametrization Geometric models are often de-

scribed by closed genus-zero surfaces ie deformed spheres For such models the sphere is the most natural parametrization domain since it does not require cutting the surface into disk(s) Hence the parametrization process becomes unconstrained Even though we may subsequently resample the surface signal onto a piecewise continuous domain these domain boundaries can be determined more conveniently and a posteriori on the sphere

the connectivity of the surface charts is used to form a domain complex that is then regularly subdivided [eg Eck et al 1995 Lee et al 1998 Kobbelt et al 1999 2 2

o e domain itself is irregular applying a texture image

to the surface requires storing a parametrization

Guskov et al 000 Lee et al 000 Wood et al 2000] Because the c nnectivity of th complex

While planar parametrization of mesh charts has been studied extensively there is relatively less work on parametrizing a mesh as a whole onto a spherical domain as reviewed in Section 31 Spherical parametrization proves to be challenging in practice for two reasons First for the algorithm to be robust it must prevent parametric ldquofoldoversrdquo and thus guarantee a 1-to-1 spherical map Second while all genus-zero surfaces are in essence sphere-shaped some can be highly deformed and creating a parametriza-tion that adequately samples all surface regions is difficult

To address these challenges our parametrization algorithm borrows techniques from Sander et al [ ] We achieve robustness using a coarse-to-fine optimization strategy and penalize undersampling using a stretch-based parametrization metric (Section 3) Intuitively stretch measures how distances in the domain get scaled onto the surface Large stretch in any direction about a surface point implies that the reconstruction of the surface signal from a domain-uniform sampling will lose high-frequency surface detail

2 Approach overview 2001 2002 Given a surface mesh M our goal is to create a geometry image I

that approximates it We first create a spherical parametrization of the surface (SrarrM) Next we similarly create a spherical parametrization (SrarrD) of a domain polyhedron D chosen to be a tetrahedron an octahedron or a cube Finally we unfold the domain into the image (DrarrI) All these maps are invertible and their composition provides a map IrarrDrarrSrarrM (see Figure 2) For remeshing we uniformly sample the domain polyhedron at the vertices of a regular n-tessellation (n+1 vertices on each domain edge) As explained in Section 41 the unfolding (IharrD) is not an isometry and we are not concerned with its metric distortion ndash it is a convenient scheme for associating the domain samples 1-to-1 with the 2D grid samples of the image

Once a spherical parametrization is obtained a number of applica-tions can operate directly on the sphere domain including shape analysis using spherical harmonics [

] compression using spherical wavelets [ ] and mesh morphing [Alexa 2002]

Funkhouser et al 2003Quicken et al 2000 Schroumlder and Sweldens 1995

Spherical remeshing Many techniques rely on sampled repre-sentations and unfortunately the only truly uniform samplings of the sphere are given by the vertices of the 5 Platonic solids (ico-sahedron being the most complex with 20 vertices)

To avoid undersampling of the surface mesh we seek to minimize the stretch of the map DrarrM We do this by minimizing stretch on the two maps DrarrS and SrarrM Both maps involve the compu-tation of a spherical parametrization as discussed next

In computer graphics spherical functions such as environment and visibility maps are often represented using cube maps [

] While this cube projection is hardware-efficient it creates a rather non-uniform spherical sampling (with stretch varying by a factor of 3) Unlike in environment maps spherical directions have no special meaning for surface parametrizations Thus we are free to construct maps that distribute samples more uniformly

3 Spherical parametrization Greene 1986 Given a triangle mesh M the problem of spherical parametriza-

tion is to form a continuous invertible map φ SrarrM from the unit sphere to the mesh The map is specified by assigning each mesh vertex v a parametrization φ-1(v) isin S Each mesh edge is mapped to a great circle arc and each mesh triangle is mapped to a spherical triangle bounded by these arcs As a sampling pattern we use the vertices of uniformly subdi-

vided regular polyhedra namely the tetrahedron octahedron and cube We explore maps with minimal stretch from these domains to the sphere (Section 42)

31 Previous work Kent et al [1992] propose several spherical parametrization schemes For general shapes they simulate a balloon inflation process but are not able to guarantee a 1-to-1 map

Geometry images The semi-regular samples that we construct over the polyhedral domains can in fact be cut and unfolded such that they correspond 1-to-1 with a regular 2D grid (Section 41) We use these unfoldings to resample our spherical parametriza-tions into geometry images (see )

Alexa [2002] uses a spring-like relaxation process The relaxation solutions may collapse to a point or experience foldovers de-pending on the starting state He demonstrates several heuristics that help the solution converge to valid maps

Figure 1The simple 2D grid structure of geometry images is ideally suited for many processing operations For instance they can be ren-dered by traversing the grids sequentially without expensive memory-gather operations (such as vertex index dereferencing or random-access texture filtering) Geometry images also facilitate compression and level-of-detail control

Grimm [ ] partitions the surface into 6 charts and maps these to faces of a cube and from there to a sphere Schemes based on a priori chart partitions constrain the spherical parametrization

Haker et al [ ] find conformal approximations of meshes over the sphere Conceptually they remove a single point from the surface harmonically map the remaining surface onto an infinite plane and finally map that infinite plane onto the sphere using stereographic projection Both maps are conformal for smooth surfaces Unfortunately many maps that are bijective and con-formal for smooth surfaces do not guarantee an embedding when applied to piecewise-linear surfaces (meshes) and sometimes produce triangle flips as in Eck et al [1995] For instance stereographic projection can flip thin obtuse triangles Another concern with conformal-like maps is scale distortion (Figure 11)

The geometry images introduced by Gu et al [2002] support surfaces of arbitrary genus by allowing an arbitrary surface cut However the cut topology must be stored explicitly for lossy decompression and it constrains level-of-detail mip-mapping In contrast our construction is specialized to genus-zero surfaces and the topology of our polyhedral cuts is fixed and simple The cuts give rise to symmetry rules at the image boundaries which enable morphing better level-of-detail control more effective compression and the creation of smooth surfaces

Sheffer et al [ ] find the angles of a spherical embedding as a constrained nonlinear system and show results for simple meshes

image I domain D sphere S surface mesh M

Gotsman et al [ ] show a nice relationship between spectral graph theory and spherical parametrization and embed simple meshes on the sphere by solving a quadratic system

Quicken et al [2000] parametrize the surface of a voxel volume onto a sphere Their nonlinear objective functional exploits the uniform quadrilateral structure of the voxel surface it seeks to equalize areas and preserve right-angles of surface elements Their scheme is not applicable to general triangle meshes Figure 2 Overview of the parametrization process

These prior schemes cannot parametrize a complex mesh robustly and with the low scale-distortion necessary for good remeshing

Gnomonic 2-slerp sym ArvoTurk-1 Buss-Fillmore Area Subdivision Stretch DrarrS (Stretch SrarrD)

DrarrS L2=031 L2=067 L2=075 L2=070 L2=040 L2=071 L2=085 (L2=080) SrarrD L2=023 L2=055 L2=073 L2=045 L2=042 L2=032 L2lt001 (L2=075) Figure 3 Comparison of spherical triangle maps for a large spherical triangle and computed stretch efficiencies in both map directions (The black curves show a uniform tessellation of the planar triangle in domain D mapped onto the sphere S)

Figure 3

32 Spherical triangle map To form a continuous parametrization φ we must define the map within each triangle interior Let the points ABC on the sphere be the parametrization of the vertices of a mesh triangle Arsquo=φ(A) Brsquo=φ(B) Crsquo=φ(C) Given a point Prsquo= αArsquo+βBrsquo+γCrsquo with bary-centric coordinates α+β+γ=1 within the mesh triangle we must define its parametrization P=φ-1(Prsquo) Any such mapping must have distortion since the spherical triangle is not developable We have explored several mappings ( ) Gnomonic It is simply spherical projection about the sphere center O That is P = (αA+βB+γC) ||αA+βB+γC|| The inverse is easily computable as spherical projection back onto the triangle 2-slerp-symmetrized Spherical linear interpolation slerp(ABα) = P finds P such that arclen(AP) arclen(PB) = α (1-α) A possible triangle interpolation scheme is to perform two consecu-tive slerps 2-slerp(ABCαβ1-α-β) = slerp(slerp(A B β(α+β)) C 1-α-β) Unfortunately this definition is not symmetric since 2-slerp(ABCαβγ) ne 2-slerp(BCAβγα) However symmetry can be obtained by averaging three 2-slerprsquos and renormalizing

2-slerp-sym(ABCαβγ) = normalize[α 2-slerp(ABCαβγ) + β 2-slerp(BCAβγα) + γ 2-slerp(CABγαβ) ]

ArvoTurk-1-symmetrized Arvo [1995] presents an area-uniform map from a square to a spherical triangle Turk [ ] presents an area-uniform map from a square to a triangle The composition of the two maps ArvoTurk-1 is an area-uniform map from a triangle to a spherical triangle Unfortunately the map is not symmetric so like 2-slerp it requires symmetrization This map is fairly expensive to compute since it involves several trigonometric evaluations

Buss-Fillmore [ ] They define barycentric combinations of spherical points A1hellipAk as least-squares minimizations of weighted geodesic distances Even for k=3 the minimization problem requires an iterative solution but it converges quickly The scheme has the property that if one takes the exponential map around the resulting point P (mapping each Ai to a point Bi in the plane tangent to the sphere at P such that ||PBi|| = arclen(PAi) and PAiBiO are coplanar) then P is the convex combination of the points Bi with the provided initial weights Consequently the spherical triangle map is easy to invert

Spherical area map In this map we obtain P such that the barycentric coordinates (αβγ) match the ratios of spherical areas of the 3 introduced spherical sub-triangles (PBC) (APC) and (ABP) We have not been able to locate a reference to this map in the literature To obtain P we first find the points DEF such that area(DBC) = αlowastarea(ABC) area(AEC) = βlowastarea(ABC) and area(ABF) = γlowastarea(ABC) Todhunter and Leathem [ ] prove that the locus of points that together with an arc XY form

spherical triangles of a given (constant) area is a small circle passing through X and Y the antipodes of X and Y Therefore P is found at the intersection of the small circles supporting (DBC) (AEC) and (ABF) The inverse map is easy to compute since it simply involves computing the ratios of the areas of the spherical sub-triangles Recursive subdivision Both the triangle and spherical triangle are recursively split using regular 1-to-4 subdivision [

] while tracking the location of the desired barycentric coordinates Since the map is not smooth neither the forward nor the inverse map has a closed-form expression

Schroumlder and Sweldens 1995

Stretch optimization For comparison we also consider the result of finely subdividing the triangle and optimizing stretch in either map direction (as described in Sections 34-35) All these maps converge to affine maps as the spherical triangle becomes small ie planar We next analyze these maps with respect to a number of desirable properties ( ) continuity at boundaries (across arbitrary adjacent triangles) arc-length pa-rametrization of boundary edges smoothness (the map should be differentiable) closed-form (analytic) expressions for both the spherical map and its inverse and distortion minimization (as measured using the stretch metric)

Table 1

Table 1 Desirable properties of a spherical triangle map

In conclusion no map exhibits all the desired properties Some maps have better stretch behavior which is important for pa-rametrizing coarse meshes However these maps are more expensive to compute and behave much like the inexpensive gnomonic map for the small triangles in fine meshes For per-formance we have chosen to use the gnomonic map for the coarse-to-fine optimization of SrarrM (Section 35) We had hoped that the good stretch behavior of the ArvoTurk-1-symmetrized map would help accelerate the coarse-to-fine optimization proc-ess but our experiments indicate that the overall compute-time convergence rate instead slows In Section 42 we also experi-ment with all these spherical triangle maps to parametrize the large domain faces in SrarrD

Boundaries Analytic Stretch Triangle mapC0 arc-len

Smooth interior DrarrS SrarrD DrarrS SrarrD

Gnomonic yes no yes yes yes poor poor2-slerp-sym yes yes yes yes no ok ok ArvoTurk-1 yes no yes yes no good goodBuss-Fillmore yes yes yes no yes good poorArea no no yes yes yes poor poorSubdivision yes yes no no no good poorStretch optim no no visually no no best best 1949

However this planar approximation to computing stretch can grossly underestimate stretch for spherical triangles with bad aspect ratio regardless of their size Specifically as a thin ldquocaprdquo spherical triangle (with one angle close to 180deg) approaches degeneracy its analytic stretch becomes infinite In contrast the stretch measured using the planar approximation approaches a finite constant since the planar triangle just rotates its plane closer to the sphere origin while maintaining a bounded aspect ratio In practice the problem reveals itself during stretch optimization as an ldquoaccordion effectrdquo

33 Review of planar-domain stretch metric Sander et al [ ] show that undersampling is directly related to the stretch of a parametrization They consider the case of a parametrization φ D rarr M (st)isinR2 rarr (xyz)isinR3 where D is a planar domain At any point (st) the singular values Γ and γ of the 3x2 Jacobian matrix Jφ = [ dφds dφdt ] represent the largest and smallest lengths obtained when mapping unit-length vectors from the domain D to the surface S ie the largest and smallest local stretch They derive two scalar norms corresponding to rms (L2) and worst-case (Linfin) stretch over all directions in the domain

Our solution is to split each spherical triangle using the perpen-dicular to the longest edge prior to subdividing it for numerical integration This split has the effect of replacing all ldquocaprdquo trian-gles with two ldquoneedlerdquo triangles (with two angles close to 90deg) A needle triangle does not suffer from the same inaccuracy since its shortest planar edge shrinks along with its spherical counterpart

( )2 212( )L s t 2γ= Γ + and ( )L s tinfin = Γ

The L2-stretch norm is L2-integrated over the surface M to obtain the stretch metric

( )2 2 2

1( ) ( ) ( )MM s t D

L M L s t dA s tA isin

= intint 35 Spherical parametrization algorithm Coarse-to-fine strategy Minimizing the stretch norm is a nonlinear optimization and we approach this problem using a coarse-to-fine multiresolution strategy as in Hormann et al [ ] and Sander et al [2002] We simplify the surface mesh M to a tetrahedron while creating a progressive mesh [Hoppe 1996] favoring triangles with good aspect ratios After mapping the base tetrahedron to the sphere we traverse the progressive mesh sequence inserting vertices on the sphere while maintaining an embedding and minimizing the stretch metric shows some example results

where ( )MdA s t ds dtγ= Γ is the differential surface area 1999For the case of a triangle mesh φ is piecewise linear and its

Jacobian Jφ is constant over each triangle Thus the integrated metric can be rewritten as a finite sum The L2 stretch efficiency is defined as where A

2 2( ) (1 ( )M DA A L M )M and AD are the areas of the surface and domain respec-

tively The stretch efficiency has an upper bound of 1 Figure 434 Spherical-domain stretch metric

Vertex insertion Each vertex split in the progressive mesh specifies a ring of vertices that will be the neighbors of the new vertex This vertex ring forms a spherical polygon The kernel of this spherical polygon is defined as the intersection of the open hemispheres defined by the polygon edges To maintain an embedding (ie avoid flipped or degenerate triangles) we place the new vertex inside this kernel Note that if the mapping is an embedding prior to the insertion the kernel cannot be empty

We extend the definition of stretch to consider a spherical pa-rametrization φ S rarr M isinS rarr (xyz)isinRv 3 We analyze the map piecewise on each spherical triangle Let (st) be a local orthonormal coordinate frame on a triangle T of M Because the spherical triangle is non-planar we find it more convenient to analyze the inverse map φ-1(st) from triangle T back to the sphere Let 1Jφ minus be the Jacobian of this inverse map Around any point φ(P) inside T 1Jφ minus provides a local linear approximation for φ-1 Consequently distances around P get stretched through map φ by a factor between 1γ and 1Γ (with Γ and γ the singular values of 1Jφ minus ) Thus the stretch over the triangle T is

1 1 1( ) ( )T

MM s t T

L T dA γisin

= + Γ

intint A s t

Vertex optimization After inserting a new vertex we optimize vertices in the neighborhood one at a time To optimize one vertex we minimize the stretch metric summed on its adjacent triangles We perform a set of great-circle searches in random directions on the sphere using bracketed parabolic minimization To prevent flipping we only consider perturbing the vertex within the kernel of its 1-ring Note that degenerate triangles are avoided since they have infinite stretch energy

where is the differential mesh triangle area ( )TMdA s t ds dt= Each time the number of vertices has increased by a constant

factor (eg 2) we sweep through all mesh vertices and optimize each one in turn More precisely we place all vertices in a prior-ity queue ordered by the amount of change in their neighborhood The process stops when the largest change is below a threshold (eg 10-3)

When Γ gtgt γ minimizing stretch alone produces oversampling along one direction Since the parametrization is not well con-strained along that direction the iso-parameter lines become ldquobunched-uprdquo and wavy (see Figure 11) To avoid this artifact we add a small regularization term ε(AM4π)p2+1(Γ)p that penal-izes inverse stretch We have found that ε=00001 and p=6 work well for all our models Unlike a conformal energy term inverse stretch only contributes in regions of oversampling

Robustness We can show by induction that our parametrization algorithm always produces an embedding We start with an embedding (since the base case is simply a tetrahedron) and we note that both types of operations (vertex insertion and vertex optimization) maintain the embedding The algorithm can only fail due to numerical precision problems and we have not experi-enced any such problems Shapiro and Tal [ ] use a similar coarse-to-fine refinement strategy to robustly map a mesh onto a convex polyhedron which could then be projected to a sphere

Since 1Jφminus and therefore Γ and γ are not constant over T we have to resort to numerical integration We subdivide each spherical triangle such that the resulting pieces are sufficiently planar (ie the length of the longest edge is bounded by a threshold) For each resulting spherical sub-triangle we can directly point-sample the Jacobian 1Jφminus

by evaluating the derivatives of the spherical triangle map Derivative computations are fairly expensive though Instead we have found it more efficient to numerically compute Jφ minus by evaluating stretch using the planar triangle spanning its vertices

Convergence As in most nonlinear problems our minimization algorithm is not guaranteed to find a global minimum The vertex optimizations always decrease the stretch energy functional and since it is positive it must converge to a local minimum

Figure 4 Spherical parametrizations Venus cow bunny David and close-up of the cow (showing snout eyes ears and horns)

4 Spherical remeshing Having parametrized a surface onto the sphere we now evaluate its signal onto one of three polyhedral domains First we show that the samples from a uniform n-tessellation of the three do-mains can be mapped to geometry images Second we present low-stretch parametrizations from the domains to the sphere

41 Domain unfolding (DrarrI) Our goal in remeshing is to represent the surface signal using a compact 2D sample grid since such a geometry image supports efficient storage access and processing Our first efforts focused on the cube domain since its 6 faces can each be represented by a square geometry image We were initially discouraged by the tetrahedron and octahedron domains since their isometric unfoldings do not yield rectangles in the plane (For example such unfoldings are used for polyhedral maps in cartography [Furuti 1997]) However we discovered that the samples of the regularly subdivided domain faces can be mapped 1-to-1 with samples of a 2D grid As shown in the tetrahedron can be unfolded into a rectangle and the octahe-dron into a square Note that the unfolded tetrahedron wraps left-to-right so its rightmost column in the geometry image is redun-dant The 1-to-1 correspondence between domain samples and images samples is illustrated in F

Figure 5

Figure 5 Cutting and unfolding the tetrahedron and octahedron The colored edges represent the domain cuts

igure 6

Figure 6 The three domains unfolded into geometry images and mapped onto the sphere (Samples lie at gridline intersections)

The remaining two Platonic solids would have more domain faces The 20 triangles of the icosahedron could be unfolded to form 10 square geometry images or 5 rectangular ones The dodecahedron seems impractical since its 12 pentagonal faces do not admit a regular sampling Since the samples in an image I really represent points on the domain D filtering and reconstruction operations over the image must consider the domain geometry Thus for domains with triangle faces (tetrahedron and octahedron) this would imply linear as opposed to bilinear reconstruction Interestingly in current graphics hardware geometry is interpo-lated linearly (using the triangle primitive) whereas textures are interpolated bi-linearly (from 2D grids) Geometry images encourage us to consider unifying the reconstruction kernels Thus for tetrahedron and octahedron domains both geometry and signals can be linearly interpolated (ie triangles with Gouraud-interpolated signal) And for the cube domain both could be bi-linearly interpolated From a signal-processing point-of-view hexagonal grids (ie formed by equilateral triangles) actually offer better reconstruction characteristics than orthogonal grids for isotropic signals [Stauton 1999] Quadrilateral geometry image If one desires a single geometry image with quadrilateral (bilinear) structure the triangle-based octahedral parametrization produces a poor sampling because the

non-isometric unfolding leads to derivative discontinuities on the domain edges (Figure 7a) As a better alternative we construct a map from a flattened octahedron to the sphere The flattened octahedron does unfold isometrically into a square and still maintains the proper spherical topology We tessellate the flat-tened octahedron and apply stretch-minimization as explained in the next section We thus obtain a map that is smooth everywhere except at the 4 image boundary midpoints and has improved stretch efficiency (see F b) igure 7

Figure 7 For a quadrilateral sampling a flattened octahedron domain unfolds isometrically onto the square yielding a smoother and more stretch-efficient parametrization

Boundary extension rules Often applications that process regular grids need to ldquohallucinaterdquo samples that lie outside the grid boundaries Traditional choices are to replicate the boundary values (ie clamping coordinates to the interior) to repeat the grid periodically or to reflect the grid across its boundaries These approaches correspond to the implied topologies of a disk (clamp-ing both axes) a cylinder (periodicity along one axis) or a torus (periodicity along both axes)

tetrahedron octahedron cube

(2n)(n+1) samples (2n+1)2 samples 6(n+1)2 samples

2n2+2 samples 4n2+2 samples 6n2+2 samples

L2(DrarrS) = 0910 L2(DrarrS) = 0969 L2(DrarrS) = 0983

image I domain D sphere S domain D sphere S

L2(IrarrS)=0839 L2(IrarrS)=0896 (a) regular octahedron (b) flattened octahedron

The boundary symmetries of our geometry images permit simple extension rules that allow continuous traversal over a spherical topology The key is to extend the grid by rotating it 180deg around the midpoints of boundaries The inset figure shows this process applied to the octahedron with the edges colored as in Figure 5 For the tetrahedron grid we apply regular periodicity along the horizontal direction and 180deg rotations in the vertical direction

tetrahedron octahedron cube flat octahedron

These grid extension rules create an infinite lattice in 2D Note that if one applies a separable symmetric filter kernel to this lattice the lattice symmetries are preserved We make use of this property for image wavelet compression (Section 6)

42 Domain spherical parametrization (DrarrS) The simplest parametrization from the domain to the sphere is a spherical projection ie using a set of gnomonic maps This is the traditional parametrization for environment cube-maps However as we saw in gnomonic projection creates high stretch distortion for large faces such as the domain faces

Figure 3

Our solution is to apply our spherical parametrization optimiza-tion of Section 3 to an n-tessellated domain Because here we seek to minimize stretch to the sphere instead of from the sphere the stretch metric is measured in the direction DrarrS and is inte-grated over the sphere (Stretch of this inverse map is easy to compute since the Jacobian singular values are reciprocals of those in the forward map) The resulting stretch-optimized maps are shown in the bottom row of Figure 6 The maps have excellent stretch efficiency from the domains to the sphere as shown in the left of Moreover the derivatives are visually continuous everywhere except at the domain vertices (which are not developable) For each domain we pre-compute the map once at a high tessellation density and save it to disk Then we can sample this map for any requested density n compares sampling the same model using all four domain types

Table 2

Table 2 L2 stretch efficiencies for DrarrS using optimization and procedural maps (Cube faces are split into 2 4 or 8 triangles)

Figure 8

Figure 8 Remeshing using the four domain mappings

5 Results and discussion Our implementation consists of two modules The first computes the surface spherical parametrization (MrarrS) This is the slowest step in our pipeline requiring about 7-25 minutes for our test models (25K-200K faces) on a 3GHz Pentium4 PC The second module subdivides the domain D and maps these samples into both the geometry image (DrarrI) and the sphere (DrarrS) To efficiently map a point from the sphere back onto the surface we locate the containing spherical triangle using a spatial data struc-ture By decomposing the parametrization into separate maps (DrarrS and MrarrS) we avoid having to split the faces of the original mesh M along the edges of the polyhedral domain D Figure 9 shows a comparison of our remeshing scheme with that of Gu et al [ ] The Gu et al parametrizations have more distortion at the cut path ends Table 3 provides quantitative comparison of spherical parametrization and remeshing using our 3 domains and the square domain of Gu et al Accuracy is meas-ured as Peak Signal to Noise Ratio PSNR = 20 log10(peakd) where peak is the bounding box diagonal and d is the symmetric rms Hausdorff error (geometric distance) between the original mesh and the remesh The tessellation number n is adjusted across domain types to approximately equalize sample sizes The PSNR numbers indicate that our spherically parametrized geome-try images have greater accuracy for sphere-like objects as was to be expected While the flexibility of the arbitrary cuts in Gu et al is advantageous for some objects with long extremities (

) our results are close The somewhat lower PSNR numbers for the cube-domain parametrizations are likely due to the heuris-tic selection of diagonals for triangulation

As an alternative we explored procedural maps using the spheri-cal triangle maps presented in Section 32 For the cube we split each cube face into 2 4 or 8 triangles prior to the triangle map-ping1 As shown in Table 2 the most efficient procedural maps (shown bold) are slightly less efficient than our optimized maps Also they generally lack derivative continuity across edges of the domain polyhedra

Figure 10

shows an inherent limitation of spherical parametriza-tion for highly deformed shapes one cannot simultaneously have both low stretch and conformality Figure 11 shows the advan-tage of the stretch metric over the more traditional conformal metric and also demonstrates the improvement due to our in-verse-stretch regularization

Procedural Maps

Stretch- optimiz Gno-

monic 2-slerp Arvo

Turk-1 Buss-

Fillmore Area Subdiv

tetra 0910 0628 0871 0846 0889 0849 0645 octa 0969 0893 0954 0943 0958 0958 0902 cube2 0859 0945 0967 0953 0932 0924 cube4 0956 0965 0966 0966 0978 0959 cube8

0983 0961 0966 0966 0966 0965 0964

flat-octa 0896 - - - - - -

Figure 14 shows more examples using the 4 domain types Note that all domains work well on any given model ndash we made no effort to select the domain per model We prefer the octahedron domain for triangle-based (linear) reconstruction and the flat-tened octahedron for quad-based (bilinear) reconstruction because they each result in a single square geometry image We explored the other two domains for research completeness The David in Figure 14 demonstrates a simple extension to our scheme that allows parametrization of surfaces with boundaries Given an initial mesh with boundaries (holes) we triangulate each hole marking the newly added faces with the attribute ldquoholerdquo We apply our spherical parametrization algorithm to this closed

1On the cube we also explored some quad-based subdivision schemes but

none of these matched the best triangle-based map

mesh but we reduce the energy computed on all hole faces by a constant factor (eg 10-6) The effect is that each hole is allowed to shrink in the parameter domain However it cannot collapse completely because the non-hole faces adjacent to the boundary do measure tangential stretch along the surface boundary The precise 3D geometry of the hole faces is unimportant since their sole purpose is effectively to prevent boundary self-intersection during optimization During remeshing we ignore samples that lie in hole faces (colored in yellow)

Geometry image Parametrization amp close-up Remesh

(a) conformal metric PSNR=312

(b) L2 stretch metric PSNR=596

(c) L2 stretch metric + L6 inverse stretch metric PSNR=581

L2 stretch efficiency DrarrM Remesh PSNR (dB) tetra octa cube [Gu] tetra

n=181 octa

n=128 cuben=104

[Gu]n=256

Venus 0864 0943 0937 0706 823 834 818 807

bunny 0673 0706 0703 0639 796 798 790 782

David 0767 0828 0822 0644 678 680 682 682

gargoyle 0568 0643 0656 0669 781 792 782 770

armadillo 0407 0454 0465 0607 712 720 720 725

horse 0349 0363 0389 0351 766 769 766 743

cow 0377 0405 0427 0582 745 752 755 780

dinosaur 0326 0360 0345 0496 731 736 732 748

Table 3 Parametrization efficiencies and remesh accuracies

Figure 11 Parametrization comparison (a) While the conformal metric better preserves angles it leads to scale distortion and thus undersampling (b) The stretch metric more uniformly distrib-utes samples but has irregular oversampling (c) Our solution is to add a fraction of inverse stretch as a regularizing term

Gu et al geometry image octahedron geometry image

6 Applications Rendering Our geometry images can be rendered as triangles by simply connecting the samples For the tetrahedron and octahe-dron domains the triangle connectivity is already provided For the cube domain we split each domain quad along the shorter diagonal as in Gu et al [2002]

Figure 9 Comparison of geometry images from Gu et al [2002] and from our octahedron domain parametrization The cut nodes in Gu et al exhibit much more distortion

Level-of-detail If one selects the tessellation number n to be a power of 2 simpler models can be obtained easily by subsampling the geometry image Gu et al demonstrated this geometric subsampling but the parametrization of their cut constrained the coarsest possible mesh usually to n=32 In contrast we can subsample all the way to n=1 obtaining coarse models with the connectivities of the domain polyhedra Here are two examples

n=1 n=2 n=4 n=8 n=16 n=32 n=64

Figure 10 This cow model has a number of narrow extremities that make spherical parametrization challenging Even at the tip of the tail our method is robust and avoids undersampling Of course parametrization anisotropy along the tail becomes un-avoidable as evidenced by the thin remesh quadrilaterals

Morphing Creating a morph between two models is easy once their spherical parametrizations are obtained Any quaternion specifies a rigid alignment of the two sphere domains Then one can intersect the two spherical triangulations to form a mutual tessellation that can interpolate the models [Alexa Alex ] a 2000

Our framework offers an alternative approach which is to use spherical remeshing to create a morphable geometry image We simply store two positions at each pixel of the geometry image and interpolate between these points as shown below

Note that the Gu et al geometry images do not support such morphs since their cut structures generally differ Our geometry images can also morph between several models or a whole space of models In contrast an approach based on mutual tessellation becomes impractical due to excessive mesh refinement To provide more control over the morph it is common to let the user specify a set of k corresponding feature points on the models The problem then becomes that of parametrization subject to constraints Praun et al [2001] present a technique involving a semi-regular domain Alexa [ ] presents a warping scheme over the sphere but the scheme does not always satisfy all con-straints Eckstein et al [2001] present a robust planar solution and perhaps their approach could be adapted to the sphere Compression We have implemented two compression methods one based on spherical wavelets [Schroumlder and Sweldens 1995] and the other on 2D image wavelets [ ] Both methods work directly on the geometry images with no explicit mesh data structures Both make use of the boundary extension rules de-scribed in Section 41 and express high-pass detail in local tangential frames estimated from the low-pass surface We follow each wavelet transform with quantization and entropy coding [Davis 1996] giving higher importance (3x) to the detail compo-nents normal to the surface [ ] We ran the Davis coder at various target bit rates to obtain the ldquospherical waveletrdquo and ldquoimage waveletrdquo data points in the rate-distortion graph of Figure

Davis 1996

Khodakovsky et al 2000

Figure 12

1000 10000 100000F i le S iz e ( b yt es)

Image Wavelets

Khodakovsky et al

Spherical Wavelets

Gu et al max(n=257n=513)

Figure 12 Rate distortion for different compression schemes applied to the bunny model Both spherical and image wavelets are computed on a geometry image of size 513x513 (n=256)

The extension rules effectively provide a smooth surface pa-rametrization (ie continuous derivatives) except at four singular points and this smoothness leads to significant compression improvements As seen in for the bunny model the resulting PSNR curve consistently outperforms the specialized mesh coder of Khodakovsky et al [2000] Figure 13 shows the reconstructed remesh at various bit rates The elegance of our approach is that it uses ordinary image wavelets (no special-case bases) applied to a regular 2D grid data structure (no pointers) Smooth subdivision Losasso et al [2003] adapt our spherical remeshing approach to construct smooth surfaces over geometry images By exploiting the extension rules of our flattened octahedron parametrization they represent a closed surface using a single bicubic B-spline patch The surface is C2 everywhere except at the four cut points where it is C1 Representing the surface as a geometry image allows it to be subdivided in the fragment shader pipeline of cur-rently available graphics hardware

33x33 geom image

closed C1 surface

For spherical wavelets we use the geometry image from the octahedron domain and the lifted butterfly wavelet basis on the octahedron As shown in Figure 12 our results are better than the compressed geometry images of Gu et al [ ] For small file sizes (le35KB) our method also outperforms the coder of Kho-dakovsky et al [ ] on the bunny model because our domain is simple and implicit

1469 bytes 2991 bytes 12065 bytes

For image wavelet coding we use the geometry image defined with the flattened octahedron domain One difficulty with using a standard image coder is that it is unaware of the boundary conti-nuity conditions and thus creates a lossy reconstructed surface with a gap along the domain cut For this reason the compression scheme of Gu et al [2002] has to subsequently fuse the boundary using boundary topology stored in a sideband This fusion results in sharp step functions that must then be diffused into the surface We are able to avoid this continuity issue altogether by modifying the image coder to use the boundary extension rules of Section 41 effectively working on an image with spherical topology This modification only involves about 100 lines of code We use symmetric image wavelet filters to preserve boundary symmetries at all image levels During reconstruction we ensure proper duplication of the boundary values at all levels of the image pyramid and as a result obtain a watertight mesh

Figure 13 Geometry compression using image wavelets (The four invisible domain cuts meet at the top of the rear right knee)

GU X GORTLER S AND HOPPE H 2002 Geometry images ACM SIGGRAPH 2002 pp 355-361

7 Summary and future work We have presented a robust algorithm for parametrizing genus-zero models onto the sphere and introduced several approaches for resampling the spherical signal onto regular domains with simple boundary symmetries We demonstrated our spherical parametrization framework on a collection of challenging models The geometry images generated using our polyhedral samplings have simple structures that benefit a number of applications including rendering level-of-detail control morphing compres-sion and hardware evaluation of smooth surfaces

GOTSMAN C GU X AND SHEFFER A 2003 Fundamentals of spherical parameterization for 3D meshes ACM SIGGRAPH 2003

GUSKOV I VIDIMČE K SWELDENS W AND SCHROumlDER P 2000 Normal meshes ACM SIGGRAPH 2000 pp 95-102

HAKER S ANGENENT S TANNENBAUM S KIKINIS R SAPIRO G AND HALLE M 2000 Conformal surface parametrization for texture mapping IEEE TVCG 6(2) pp 181-189

HOPPE H 1996 Progressive meshes ACM SIGGRAPH 96 pp 99-108 HORMANN K GREINER G AND CAMPAGNA S 1999 Hierarchical

parametrization of triangulated surfaces Vision Modeling and Visu-alization 1999 pp 219-226

There are a number of areas for future work A more thorough investigation and comparison of compression approaches for spherically parametrized models

KENT J CARLSON W AND PARENT R 1992 Shape transformation for polyhedral objects ACM SIGGRAPH 92 pp 47-54

Treatment of the parametric singularities at the domain vertices to allow more general signal-processing to be applied to the surfaces such as general convolution

KHODAKOVSKY A SCHROumlDER P AND SWELDENS W 2000 Progressive geometry compression ACM SIGGRAPH 2000 pp 271-278

Use of fine-to-coarse integrated metric tensors to accelerate spherical parametrization and to enable signal-specialized pa-rametrization as in Sander et al [ ] 2002

SANDER P GORTLER S SNYDER J AND HOPPE H 2002 Signal-specialized parametrization Eurographics Workshop on Rendering 2002

KOBBELT L VORSATZ J LABSIK U AND SEIDEL H-P 1999 A shrink wrapping approach to remeshing polygonal surfaces Eurographics 1999 pp 119-130

LEE A SWELDENS W SCHROumlDER P COWSAR L AND DOBKIN D 1998 MAPS Multiresolution adaptive parametrization of surfaces ACM SIGGRAPH 98 pp 95-104

Direct minimization of stretch across DrarrM to account for slight non-uniformity of DrarrS map perhaps by modulating the stretch metric tensor during SrarrM optimization

LEE A MORETON H AND HOPPE H 2000 Displaced subdivision surfaces ACM SIGGRAPH 2000 pp 85-94 Consistent spherical parametrizations among several models

(ie with feature correspondences) LEacuteVY B PETITJEAN S RAY N AND MAILLOT J 2002 Least squares conformal maps for automatic texture atlas generation ACM SIG-GRAPH 2002 pp 362-371

Acknowledgments We thank the Digital Michelangelo Project at Stanford University for the David surface model

LOSASSO F HOPPE H SCHAEFER S AND WARREN J 2003 Smooth geometry images Submitted for publication

MAILLOT J YAHIA H AND VERROUST A 1993 Interactive texture mapping ACM SIGGRAPH 93 pp 27-34

References ALEXA M 2000 Merging polyhedral shapes with scattered features The

Visual Computer 16(1) pp 26-37 PRAUN E SWELDENS W AND SCHROumlDER P 2001 Consistent mesh

parametrizations ACM SIGGRAPH 2001 pp 179-184 ALEXA M 2002 Recent advances in mesh morphing Computer Graph-

ics Forum 21(2) pp 173-196 QUICKEN M BRECHBUumlHLER C HUG J BLATTMANN H SZEacuteKELY G

2000 Parametrization of closed surfaces for parametric surface de-scription CVPR 2000 pp 354-360 ARVO J 1995 Stratified sampling of spherical triangles ACM SIG-

GRAPH 95 pp 437-438 SANDER P SNYDER J GORTLER S AND HOPPE H 2001 Texture mapping progressive meshes ACM SIGGRAPH 2001 pp 409-416 BUSS S AND FILLMORE J 2001 Spherical averages and applications to

spherical splines and interpolation ACM Transactions on Graphics 20(2) pp 95-126

CIGNONI P MONTANI C ROCCHINI C AND SCOPIGNO R 1998 A general method for recovering attribute values on simplified meshes IEEE Visualization 1998 pp 59-66

SCHROumlDER P AND SWELDENS W 1995 Spherical wavelets Efficiently representing functions on the sphere ACM SIGGRAPH 95 pp 161-172 COHEN J OLANO M AND MANOCHA D 1998 Appearance-preserving

simplification ACM SIGGRAPH 98 pp 115-122 SHAPIRO A AND TAL A 1998 Polygon realization for shape transforma-tion The Visual Computer 14 (8-9) pp 429-444 DAVIS G 1996 Wavelet image compression construction kit

httpwwwgeoffdavisnetdartmouthwaveletwavelethtml SHEFFER A GOTSMAN C AND DYN N 2003 Robust spherical parameterization of triangular meshes 4th Israel-Korea Bi-National Conf on Geometric Modeling and Computer Graphics pp 94-99

ECK M DEROSE T DUCHAMP T HOPPE H LOUNSBERY M AND STUETZLE W 1995 Multiresolution analysis of arbitrary meshes ACM SIGGRAPH 95 pp 173-182 SHEFFER A AND HART J 2002 Seamster Inconspicuous low-distortion

texture seam layout IEEE Visualization 2002 pp 291-298 ECKSTEIN I SURAZHSKY V AND GOTSMAN C 2001 Texture mapping with hard constraints Eurographics 2001 pp 95-104 SORKINE O COHEN-OR D GOLDENTHAL R AND LISCHINSKI D 2002

Bounded-distortion piecewise mesh parametrization IEEE Visualiza-tion 2002

FLOATER M 1997 Parametrization and smooth approximation of surface triangulations CAGD 14(3) pp 231-250

FUNKHOUSER T MIN P KAZHDAN M CHEN J HALDERMAN A DOBKIN D AND JACOBS D 2003 A search engine for 3D models ACM Transactions on Graphics 22(1) January 2003 pp 83-105

STAUNTON R 1999 Hexagonal sampling in image processing In Advances in imaging and electron physics Vol 107 Academic Press

TURK G 1990 Generating random points in triangles Graphics Gems Academic Press pp 649-650 FURUTI C 1997 httpwwwprogonoscomfuruti

GREENE N 1986 Environment mapping and other applications IEEE Computer Graphics and Applications 6(11)

TODHUNTER I AND LEATHEM JG 1949 Spherical Trigonometry Macmillan and Co Limited

GRIMM C 2002 Simple manifolds for surface modeling and parametri-zation Shape Modeling International 2002

WOOD Z DESBRUN M SCHROumlDER P AND BREEN D 2000 Semi-regular mesh extraction from volumes IEEE Visualization 2000 pp 275-282

tetrahedron parametrization geometry image (362x182) flat-shaded remesh (65524 unique vertices)

octahedron parametrization geometry image (257x257) flat-shaded remesh (65538 unique vertices)

cube parametrization geometry image (6x105x105) flat-shaded remesh (64523 non-hole vertices)

flat octahedron parametrization geometry image (257x257) flat-shaded remesh (65538 unique vertices)

Figure 14 Results of spherical parametrization and remeshing using all four domain types For visualization the geometry images in the middle column are shown shaded using two antipodal lights (Shading is based solely on the remesh ndash there are no stored normals)

Introduction

Approach overview

Spherical parametrization

Previous work

Spherical triangle map

Review of planar-domain stretch metric

Spherical-domain stretch metric

Spherical parametrization algorithm

Spherical remeshing

Domain unfolding (DI)

Domain spherical parametrization (DS)

Results and discussion

Applications

Summary and future work

Acknowledgments

References